Gromov boundary

The Cayley graph of a free group with two generators. This is a hyperbolic group whose Gromov boundary is a Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs.

The (6,4,2) triangular hyperbolic tiling. The triangle group corresponding to this tiling has a circle as its Gromov boundary.

In mathematics, the Gromov boundary of a delta-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually, the Gromov boundary is the set of all points at infinity. For instance, the Gromov boundary of the real line is two points, corresponding to positive and negative infinity.

Properties of the Gromov boundary

The Gromov boundary has several important properties:

• The Gromov boundary of a hyperbolic group is compact.
• If a group $G$ acts geometrically on a delta-hyperbolic space, then $G$ is is hyperbolic group and $G$ and $X$ have homeomorphic Gromov boundaries.
• The Gromov boundary of a tree is a Cantor set.

Cannon's Conjecture

Main article: Cannon's Conjecture

Cannon's conjecture concerns the classification of groups with a 2-sphere at infinity:

Cannon's conjecture: Every Gromov hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.

The analog to this conjecture is known to be true for 1-spheres and false for spheres of all dimension greater than 2.